The doomsday argument is sound. It's controversial with the assumptions about population growth, which plain don't make sense—I don't claim to understand what happens, but I think that an argument would need to be made for the entire population basically vanishing when it hits the cap. There are certainly scenarios that would lead to that (e.g. nuclear war over resources could plausibly destroy statistically significant human populations enough to allow for extinction), but again, it requires argument.

Or another way, you at least need to argue AGAINST rational rationing of existing resources to make calculating potential populations over all time quite difficult and run up against heat death calculations.

That said, I love both these thought experiments because they highlight how hard it is to figure out population growth with (theoretically) rational populations.